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Eckart-Young theorem
For a matrix $A \in \mathbb{R}^{m \times n}$, it must have an SVD decomposition $A=U \Sigma V^{T}$ with singular values $\sigma_ {1} \geq \sigma_ {2} \geq \ldots \geq \sigma_ {p} \geq 0$ and $p=\min {m, n}$ . Using $\left|U^{T} A V\right|=|\Sigma|...…
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Singular Value Decomposition (SVD)
Singular Value Decomposition (SVD) is the most important decomposition method in linear algebra and has a deep connection with Principle Component Analysis (PCA) in machine learning. SVD says any matrix $A \in \mathbb{R}^{m \times n}$ of rank $r$ ...…
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DistMesh 使用教程
1、主要函数 函数 distmesh2d( )DISTMESH2D 2-D Mesh Generator using Distance Functions.[P,T]=DISTMESH2D(FD,FH,H0,BBOX,PFIX,FPARAMS)% P: Node positions (Nx2), 网格点的x,y坐标% T: Triangle indices (NTx3), 三角形单元的三个顶点索引% FD: Di...…
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Gradient,Divergence and Curl
1、哈密尔顿算子: $\nabla$ -nabla在介绍梯度等概念之前，首先引入CFD非常常见的运算符之一: $\nabla$, 它是某一物理量在三个坐标方向的偏导数的矢量和, 定义如下:\[\nabla=\frac{\partial}{\partial x} \mathbf{i}+\frac{\partial}{\partial y} \mathbf{j}+\frac{\partial}{\partial z} \mathbf{k}\]2、梯度（Gradient)当 $\nabla$ 作...…
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Model reduction techniques for fast blood flow simulation
AbstractIn this paper, we propose a new model reduction technique aimed at real-time blood flow simulations on a given family of geometrical shapes of arterial vessels. Our approach is based on the combination of a low-dimensional shape parametriz...…
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Maxwell方程的DG算法数值实现
1. 原理简介一维Maxwell方程组形式如下\[\left\{ \begin{aligned}\varepsilon(x) \frac{\partial E}{\partial t}=-\frac{\partial H}{\partial x}\\ \mu(x) \frac{\partial H}{\partial t}=-\frac{\partial E}{\partial x}\end{aligned}\right. \label{eq1} \tag{1}\]其中 $(E,H)$ 分...…
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Numerical linear algebra
1. Gram-Schmidt正交化的修正策略(标准 Gram-Schmidt 正交化) 给定 $n \times m(m \leqslant$$n)$ 阶列满秩矩阵 $\boldsymbol{X}=\left[\boldsymbol{x}_ {1}, \boldsymbol{x}_ {2}, \cdots, \boldsymbol{x}_ {m}\right]$, 本算法产 生 $n \times m$ 阶正交矩阵 $\boldsymbol{Q}=\left[\boldsymbol{q}...…
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Transfinite maps
AbstractWe present a method to generate a non-affine transfinite map from a given reference domain to a family of deformed domains. The map is a generalization of the Gordon–Hall transfinite interpolation approach. It is defined globally over the ...…
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Parametrized PDEs
1.Parametrized partial differential equations notation meaning $\Omega$ $\Omega \subset \mathbb{R}^{2}$, a fixed, parameter-independent domain $\widetilde{\Omega}$ $\widetilde{\Omega}=\wideti...…
